Beyond the Neutron Drip-Line: The Unbound Oxygen Isotopes O and O
The very neutron-rich oxygen isotopes O and O are investigated experimentally and theoretically. The unbound states are populated in an experiment performed at the R3B-LAND setup at GSI via proton-knockout reactions from F and F at relativistic energies around 442 MeV/nucleon and 414 MeV/nucleon, respectively. From the kinematically complete measurement of the decay into O plus one or two neutrons, the O ground-state energy and width are determined, and upper limits for the O ground-state energy and lifetime are extracted. In addition, the results provide indications for an excited state in O at around 4 MeV. The experimental findings are compared to theoretical shell-model calculations based on chiral two and three-nucleon (3N) forces, including for the first time residual 3N forces, which are shown to be amplified as valence neutrons are added.
pacs:21.10.-k, 25.60.t, 27.30.+t, 29.30.Hs
present address: ]TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada
Understanding the properties of nuclei with extreme neutron-to-proton ratios presents a major challenge for rare-isotope beam experiments and nuclear theory. Nuclei located at and beyond the neutron dripline play a crucial role in this endeavor. Experimentally, the neutron dripline has been established up to oxygen Langevin et al. (1985); Guillemaud-Mueller et al. (1990); Tarasov et al. (1997) with O being the last bound isotope, while it extends considerably further in fluorine Sakurai et al. (1999). Recently, it has been shown that the anomalous behavior in the oxygen isotopes is due to the impact of three-nucleon (3N) forces, which provide repulsive contributions to the interactions of valence neutrons Otsuka et al. (2010), connecting the frontier of neutron-rich nuclei to the theoretical developments of nuclear forces.
Another striking feature in the oxygen isotopic chain is the doubly-magic nature of O and O Hoffman et al. (2008); Kanungo et al. (2009); Tshoo et al. (2012); Thirolf et al. (2000); Brown and Richter (2005); Becheva et al. (2006) in strong contrast to the lighter elements, where the dripline is marked by nuclei exhibiting a loosely-bound halo structure. The neutron-rich oxygen isotopes also provide interesting insights, when viewed coming from their stable isotones. As protons are removed, the attractive contribution from the proton-neutron tensor force decreases, thus, opening up the neutron shell gap for oxygen Otsuka et al. (2005), while reducing the gap at which is very prominent in stable nuclei.
How the structure evolves beyond O towards is thus of central interest. Currently, O and O are at the limit of experimental availability. For the former isotope, the ground-state resonance energy and width have been reported Hoffman et al. (2008). For the latter, its position has been measured previously Lunderberg et al. (2012). Taking advantage of the large angular acceptance for neutrons in the R3B-LAND experiment Aumann (2007); R3B-Collaboration (2005), we investigate the unbound isotopes O and O in an extended energy range with an essentially constant efficiency up to a decay energy of 4 MeV and 8 MeV, respectively.
It has been speculated that the unbound isotopes O and O might have a rather long lifetime, which would constitute the first example of neutron radioactivity Grigorenko et al. (2011). Our present result establishes an upper limit for the lifetime of the O ground state. We then combine the experimental investigation with theoretical calculations based on chiral two-nucleon (NN) and 3N forces, where we focus on the increasing contribution from residual three-neutron forces as neutrons are added.
The experiment was carried out at the GSI Helmholtzzentrum in Darmstadt using the R3B-LAND reaction setup. Beams of light neutron-rich nuclei were produced by fragmentation reactions of a 490 MeV/nucleon Ar primary beam in a 4 g/cm Be target. Ions with a magnetic rigidity of 9.88(%) Tm corresponding to an ratio of about 3 were selected by the fragment separator FRS Geissel et al. (1992) and transported to the experimental area. Energy-loss and time-of-flight measurements allowed for the identification of the incoming ions on an event-by-event basis. The beam cocktail contained F ions (1%), which were selected to populate the unbound states in O via one-proton knockout reactions. The energies (intensities) of the F and F beams were 442 MeV/nucleon and 414 MeV/nucleon (1 Hz and 0.3 Hz), respectively. Different secondary targets (922 mg/cm CH, 935 mg/cm C, and 2145 mg/cm Pb) were used, and all shown spectra display the contributions from all targets. The target was surrounded by the 4 Crystal Ball detector Metag et al. (1983) consisting of 160 NaI crystals for detecting photons and light particles emitted at laboratory angles larger than relative to the beam axis. Position and energy loss of the beam and fragments behind the target were measured by two silicon-strip detectors before deflection in the large-gap dipole magnet ALADIN. Two further position measurements behind the magnet using scintillating fiber detectors Cub et al. (1998); Mahata et al. (2009) allowed for tracking of the ions through the dipole field. Together with time-of-flight and energy-loss measurements, this provides the magnetic rigidity and atomic number, and thus the mass of the fragments.
Neutrons from the decay of unbound states were detected at a distance of around 12 m downstream of the target by the LAND neutron detector Blaich et al. (1992) with an efficiency of 92% for single neutrons and with an angular acceptance of 79 mrad around the beam axis. A similar experimental setup and analysis scheme is described in Ref. Palit et al. (2003) in more detail.
Iii Analysis and Results
iii.1 O ground-state resonance
From the measurements of the momenta of outgoing fragments and neutrons, the two and three-body relative-energy () spectra are reconstructed for one and two-neutron events. Fig. 1 shows the O+ spectrum after proton removal from F. A prominent peak structure is visible at about 700 keV corresponding to the ground-state resonance of O. The position and width of the resonance have been extracted by fitting a Breit-Wigner distribution with an energy-dependent width using the following function Lane and Thomas (1958)
with the resonance shift set to zero and the width given by with the reduced width amplitude and the penetration factor . The penetration factor (taken from Ref. Bohr and Mottelson (1969)) depends on the channel radius , the energy and angular momentum . As the distribution was found to be insensitive to changes in between 3.5 and 6 fm, a channel radius of 4 fm was used. An angular momentum of is used, since the additional neutron of O compared to O is most likely in the orbital.
This distribution has been convoluted with the experimental response. A non-resonant background (BG) has been modeled as the product of an error and of an exponential function:
with free paramters , and . The sum of the convoluted Breit-Wigner and background functions was used to fit the experimental data.
For the minimization procedure, Pearson’s chi-square method Baker and Cousins (1984), using errors of the parent distribution according to a Poisson probability distribution, has been used, as the usual method with errors estimated from the number of counts gives inaccurate results in case of low statistics. The extracted position (width) of keV ( keV) is in agreement with the previously reported value Hoffman et al. (2008) within 1 (2), see Table 1. Our result is in agreement with a single-particle width keV calculated for a pure -state. The relatively large experimental error on the width is due to the instrumental energy resolution which dominates the apparent width, see Fig. 2. For further discussion, results from literature and the present result were averaged according to Ref. Barlow (2004) resulting in keV and . These values are compared in the lower panel of Fig. 3 to the expected widths and lifetimes as a function of resonance energy for different neutron angular momenta (adopted from Fig. 2 (b) of Ref. Grigorenko et al. (2011)). We note that the averaged width is close to the estimated value for a -state as given in Ref. Grigorenko et al. (2011).
iii.2 O ground-state resonance
The experimental spectrum for O is shown in Fig. 4, where O has been detected in coincidence with two neutrons. Two groups of events are observed: below 1 MeV and around 4 MeV. The experimental response, indicated by the red curve, is rather constant over the displayed energy region, but exhibits a steep fall-off for energies below 500 keV. For such small relative energies, the neutrons are not well separated in space and time when interacting in the detector and can thus hardly be distinguished from events. The effect can be seen quantitatively in the two-dimensional response matrices shown in Fig. 5. The energy reconstructed from the simulation is plotted versus the generated one in the upper panel, showing a band along the diagonal with a width reflecting the instrumental resolution, which is shown in Fig. 2. For low generated relative energies (100 keV), the events spread to a higher reconstructed energy and are, in addition, reconstructed as events with a large probability. This can be seen in the lower panel of Fig. 5, which shows the reconstructed O+ relative energy spectrum for the events falsely identified as events (either due to the effect discussed above at low relative energies, or due to limited coverage of the detector for high relative energy). The simulation is based on measured real events from deuteron breakup reactions. The events are generated by overlaying the shower patterns of secondary particles from these measured events according to the simulated positions on the neutron detector and are then analyzed in the same way as the experimental data Boretzky et al. (2003). The generated response matrices thus do not rely on a simulation of the reactions in the neutron detection process.
The effect discussed above is clearly visible in the O data of Fig. 4 and Fig. 6. In this channel (Fig. 6), the accumulation of events at very low energy in the first bin is compelling. This feature is not present in the events measured in the proton knockout from F, as seen in Fig. 1. The events in the first bin of the O+ spectrum are the characteristic fingerprint of a very low-lying state in O. The events with energies above 0.2 MeV can be attributed to several processes, such as a possible direct two-nucleon knockout reaction. In particular, the events between 0.2 MeV and 2 MeV could result from knockout from F populating the O ground-state resonance, shown in Fig. 1. At higher energies, -decay events can contribute due to the limited detector acceptance. According to the simulation, three counts are expected in the spectrum between 0 and 5 MeV stemming from the resonance-like structure in the channel at 4 MeV. In addition, knockout of more deeply bound protons is expected, yielding higher excitation energies in O, which will appear as a broad background in the spectrum.
We have performed a simultaneous statistical analysis of and coincidences, starting with the hypothesis of a low-lying state in O. The dotted histograms in Fig. 4 (upper panel) and Fig. 6 display the most probable result yielding a position for the O ground state of keV. Again, the minimization using Poisson distributed errors of the response function has been used. The response functions have been used to approximate the line shape.
Another method Bevington and Robinson (1992), which is independent of the binning of the experimental data, is to use the measured relative energy in each event in conjunction with the transposed response matrix to calculate the probability distribution of the true energy (Bayesian approach with uniform prior). The resulting Bayesian interval runs from 0 to 40/120 keV at a 68%/95% confidence level (c.l.). Again, both the and data are considered simultaneously.
We cannot exclude a very small value close to zero for position and width from the energy measurement alone, potentially leading to a rather long-lived O ground state, which would constitute the first case of neutron radioactivity as speculated in Ref. Grigorenko et al. (2011): As can be seen from Fig. 3, the lifetime for a -state could reach seconds for a resonance position well below 1 keV. Such a long lifetime, however, can be excluded from the fragment measurement. The distance of the target to the middle of the dipole magnet measures 256 cm, corresponding to a flight time of 11.8 ns. If O would decay after that time, a fragment mass greater than 24 would be reconstructed by the tracking procedure. Also, no neutron coincidences should be observed in that case, since the fragment is bent by 7 degree after passing half of the dipole field.
In order to obtain an upper limit on the number of events belonging to the previously described event class ( and no neutron in coincidence), the fragment mass spectrum has been inspected under the condition that no neutron is detected in coincidence, i.e., the neutron detector was used as a veto (the efficiency to detect a neutron at forward direction is 92%). The reaction trigger was provided by the Crystal Ball (CB), since for the case of proton knockout reactions, the knocked-out proton is detected at large angles with high probability in the CB. This is not only the case for FO quasi-free reactions on the hydrogen in the CH target, but also for knockout reactions induced by composite targets. The resulting fragment-mass distribution for incoming F and outgoing oxygen isotopes () is compared in Fig. 7 for the described trigger condition with the distribution obtained with neutron coincidences. As can be seen by comparing the two distributions in Fig. 7, there are only very few events observed without coincident neutrons, which can be explained by the neutron detection efficiency of 92% and by a small amount of background events. Only one event appears above the O mass, which could be attributed to background (since there are also few events in that mass range with neutron coincidences). This one count, however, provides an upper limit on the survival probability of O. Assuming a Poisson distribution, an expectation value of 4.9 would yield a probability of % (2) to detect more than one event. From the analysis of the energy spectra discussed above (see Fig. 4), the number of produced O in the ground state can be estimated. From the initial number of O, the upper limit ns; of surviving O ions and the corresponding time of flight, we obtain an upper limit for the lifetime of 5.7 ns at a 95% confidence level. The upper limits for the energy of the state and for its lifetime are shown in Fig. 3, delimiting the shaded area as the allowed region, which overlaps with the calculated values for a pure -state. A more complex O ground-state configuration, however, cannot be excluded.
|-||Hoffman et al. (2008)|
|742||NN+3N + residual 3N||this work|
|1301/1303||USDA/B||Brown and Richter (2006)|
|1002||-||-||Volya and Zelevinsky (2006)|
|O(g.s.)||11168%/95% c.l.||222from lifetime estimate, see text and Fig. 3||33395% c.l.||this work|
|-||-||Lunderberg et al. (2012)|
|40||NN+3N + residual 3N||this work|
|501/356||USDA/B||Brown and Richter (2006)|
|21||0.02||-||Volya and Zelevinsky (2006)|
It would therefore be very interesting to determine the lifetime experimentally more precisely in order to gain insights on the structure of O. Such a measurement would be rather straightforward using a similar method as described here, but placing the target directly in front of the dipole. The decay curve of O would translate into a fragment-mass distribution depending on the decay position. In addition, the neutrons will be detected not only at zero degree, but also in the bending direction, again directly reflecting the decay curve. With the intensities available, e.g., at the RIBF facility at RIKEN, a precise value could be obtained from such an experiment with the SAMURAI setup.
iii.3 Comparison with shell-model calculations
We compare the ground-state energies of O to theoretical shell-model calculations based on chiral effective field theory potentials combined with renormalization-group methods to evolve nuclear forces to low-momentum interactions Bogner et al. (2010). Our results are based on chiral NN and 3N forces, where the single-particle energies and two-body interactions of valence neutrons on top of a O core are calculated following Refs. Holt et al. (2011); Gallant et al. (2012) without adjustments. Fig. 8 shows the predicted ground-state energies obtained by full diagonalization in the valence shell .
In addition to the 3N contribution to single-particle energies and two-body interactions of valence neutrons, obtained by normal-ordering, a third contribution is given by the weaker residual three-valence-neutron forces. Here, we focus on the relative contribution from these residual forces, which become more important with increasing neutron number along isotopic chains Friman and Schwenk (2011). In order to quantify the relative contribution from residual 3N forces, we use the wavefunction and calculate the correction . The repulsive contribution increases from 0.1 to 0.4 MeV for O, highlighted by the arrows in Fig. 8. Because the ground states of O are very narrow, thus quasi-bound, they can be treated fairly well in a bound-state approximation. The remarkable agreement with experiment should, however, be considered with caution, as the continuum needs to be included. The expected contribution is about 200 keV to both O Hagen et al. (2012), so relative energy differences will be smaller. We have also explored uncertainties in the calculation of the single-particle energies and valence interactions, which would increase the ground-state energy relative to O for O by MeV. In Table 1, we also compare with the phenomenological USDA/B interactions Brown and Richter (2006), which predict too high energies for O. Better agreement is found in Table 1 for the continuum shell model Volya and Zelevinsky (2006) and in recent coupled-cluster calculations with 0.4 MeV and 0.1 MeV for O and O, respectively Hagen et al. (2012).
iii.4 O excited state
We now turn back to the group of events around 4 MeV in Fig. 4, which we interpret as resulting from the population of an excited state in O.
Assuming first that all events above the ground state could be explained by a non-resonant background (Eq. 2), this yields a fit to the data in the 1 to 8 MeV range as shown in Fig. 4 (bottom) by the blue dotted curve. The probability that the observed number of counts in the 4–5 MeV bin are explained by this fit — yielding an expectation value of 1.60 — is less than 0.018. In turn, the probability that the accumulation of events in this energy region corresponds to a peak is larger than 98% ( significance) even in a worst case scenario.
The use of a more realistic description of the data which includes three contributions (one from the O ground state resonance, another from an excited state around 4 MeV, and a third contribution for the background) results in the fit to the data as displayed by the black solid line in Fig. 4 (bottom), which represents the fit function integrated over the experimental bin width. The contribution of the background to the total fit is shown by the red dashed line, resulting in a probability of 1.2 for the events between 4 and 5 MeV to belong to the background, which is equivalent to a significance of for a peak structure.
The theoretical calculations based on chiral NN and 3N forces predict a first excited state in O at 1.6 MeV above the O ground state. It is found at 1.9/2.1 MeV for USDA/B interactions and at 1.8 MeV for Ref. Volya and Zelevinsky (2006). Experimentally, the events at 4 MeV in the three-body energy spectrum (Fig. 4) provide an indication for an excited state in O, with a most probable energy of keV. As for the ground-state resonance, the response functions have been used to approximate the line shape. The minimum corresponds to the energy bin from 4200 to 4250 keV, which is shown in Fig. 4 including the experimental response. The errors have been determined from the -distribution using the interval given by .
A likely candidate would be a proton-hole state populated after knockout from the F shell (rather than from the valence orbital). In order to investigate proton (and neutron) cross-shell excitations, we have carried out calculations in the space using the WBP interaction Warburton et al. (1990), for which the first and second energies are located at 2.3 and 5.4 MeV, respectively. The first state with a proton excitation component from to is a state at a higher energy of 5.4 MeV. Its proton contribution is also mixed with neutron excitations and considerably weaker than for the corresponding state at 5.0 MeV in O. Note that O is bound by 1.0 MeV for the WBP interaction, such that 1 MeV uncertainties are possible and the continuum should be included. The lowest negative parity states predicted by the WBP interaction are a quartet of neutron excitations from to at 3.7, 4.1, 4.5, 4.9 MeV (with centroid at 4.1 MeV). A conclusion on the character of the resonance-like structure around 4 MeV in the experimental spectrum cannot be drawn from the present study. A high-statistics experiment, which would allow for an investigation of the correlations in the three-body decay, is necessary to shed light on the structure and quantum numbers of this state.
Iv comment on the O lifetime
During the final stages of the refereeing process two Letters have been published in PRL discussing the lifetime of O, one experimental Kohley et al. (2013) and the other theoretical Grigorenko et al. (2013). In the experimental work by the NSCL-MoNA group, a half-life (stat)(syst) ps has been extracted, corresponding to a lifetime of 6.5 ps. At an 82% confidence level, a finite lifetime of the O ground state is claimed Kohley et al. (2013), suggesting the possibility of two-neutron radioactivity. Our upper limit of 5.7 ns (95% c. l.) is in agreement with this finding. Even the combination of both results does not allow for a firm conclusion ( signal) on the possibility of neutron radioactivity. It is therefore of utmost importance to perform a dedicated and optimized experiment with good statistics to measure the lifetime of O. With the method we have proposed in this paper, namely to measure the decay of O in a magnetic dipole field, it will be difficult, however, to measure a lifetime shorter than 10–100 ps. The sensitivity range depends on the field strength and in particular on the angular resolution for the neutron detection, limiting the sensitivity with present detectors to around 100 ps, and to around 10 ps at the future R3B facility at FAIR. A more elaborated discussion of the method proposed in section III.B of this paper has been published in the meantime by Thoennessen et al. Thoennessen et al. (2013), together with a more detailed discussion of the method used by Kohley et al. Kohley et al. (2013).
In summary, we have investigated the ground-state energy, width, and lifetime of the unbound oxygen isotopes O and O. Our results are in very good agreement with theoretical shell-model calculations based on chiral NN and 3N forces, where the ground-state energy of these extremely neutron-rich isotopes becomes increasingly sensitive to 3N forces among the valence neutrons. The O ground state is unbound by less than 120 keV, and our measurement provides a limit on the lifetime of 5.7 ns (both at 95% c.l.). We also obtained indications for an excited state of O located at about 4 MeV. Different possibilities for the nature of this state exist, making it an exciting case for future calculations and experiments with higher statistics.
Acknowledgements.We thank B. A. Brown for helpful discussions on the WBP interaction. This work was supported by the Helmholtz International Center for FAIR within the framework of the LOEWE program launched by the state of Hesse, by the Helmholtz Alliance Program of the Helmholtz Association, contract HA216/EMMI “Extremes of Density and Temperature: Cosmic Matter in the Laboratory”, by the GSI-TU Darmstadt Cooperation agreement, by the BMBF under Contracts No. 06DA70471, 06DA9040I, 06MT238, by the DFG cluster of excellence Origin and Structure of the Universe, by the US DOE Grants DE-FC02-07ER41457, DE-FG02-96ER40963, via the GSI-RuG/KVI collaboration agreement, and by the Portuguese FCT, project PTDC/FIS/103902/2008.
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